Shy couplings, CAT(0) spaces, and the lion and man
Bramson, Maury, 1951-, Burdzy, K. (Krzysztof) and Kendall, Wilfrid S.. (2013) Shy couplings, CAT(0) spaces, and the lion and man. Annals of Applied Probability, 41 (2). pp. 744-784. ISSN 1050-5164
WRAP_Kendall_BramsonBurdzyKendall-2010.pdf - Draft Version - Requires a PDF viewer such as GSview, Xpdf or Adobe Acrobat Reader
Official URL: http://dx.doi.org/10.1214/11-AOP723
Two random processes X and Y on a metric space are said to be ε-shy coupled if there
is positive probability of them staying at least a positive distance ε apart from each other forever.
Interest in the literature centres on nonexistence results subject to topological and geometric conditions; motivation arises from the desire to gain a better understanding of probabilistic coupling.
Previous non-existence results for co-adapted shy coupling of reflected Brownian motion required
convexity conditions; we remove these conditions by showing the non-existence of shy co-adapted
couplings of reflecting Brownian motion in any bounded CAT(0) domain with boundary satisfying uniform exterior sphere and interior cone conditions, for example, simply-connected bounded
planar domains with C2 boundary.
The proof uses a Cameron-Martin-Girsanov argument, together with a continuity property of
the Skorokhod transformation and properties of the intrinsic metric of the domain. To this end, a
generalization of Gauss' Lemma is established that shows differentiability of the intrinsic distance
function for closures of CAT(0) domains with boundaries satisfying uniform exterior sphere and
interior cone conditions. By this means, the shy coupling question is converted into a Lion and
Man pursuit-evasion problem.
|Item Type:||Journal Article|
|Subjects:||Q Science > QA Mathematics|
|Divisions:||Faculty of Science > Statistics|
|Library of Congress Subject Headings (LCSH):||Brownian motion processes|
|Journal or Publication Title:||Annals of Applied Probability|
|Publisher:||Institute of Mathematical Statistics|
|Official Date:||March 2013|
|Page Range:||pp. 744-784|
|Access rights to Published version:||Restricted or Subscription Access|
|Funder:||National Science Foundation (U.S.) (NSF), Poland. Ministerstwo Nauki i Szkolnictwa Wyższego [Ministry of Science and Higher Education] (MNiSW)|
|Grant number:||CCF-0729537 (NSF), DMS-0906743 (NSF), N N201 397137 (MNiSW)|
Alexander, S. B., R. L. Bishop, and R. Ghrist (2006). Pursuit and evasion in non-convex domains
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